Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,8,5,20)(3,12,16,22,13,27,15,9)(4,17,18,23,19,25,21,10)(14,24,29,31,28,32,30,26);(2,8)(3,13)(5,6)(9,22)(14,28)(17,25)(18,21)(26,31);(1,3,5,15,7,13,6,16)(2,9,11,12,8,22,20,27)(4,14,18,29,19,28,21,30)(10,24,25,26,23,32,17,31);(1,4)(2,10,8,23)(3,14)(5,18)(6,21)(7,19)(9,24,22,32)(11,25,20,17)(12,26,27,31)(13,28)(15,29)(16,30)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319831, #72320044, #72320069, … |
| maximal normal subgroup of | #72319831, #72320044, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291532);